VECTORS

1
VECTORS
THE MAGIC OF VECTOR MATH
2
2 Types of Quantities

• Scalars
• Just a Value
• This Value is called a Magnitude
• Vectors

3
VECTORS

• Quantities that have ?
• MAGNITUDE
  (size or value)
• AND
• DIRECTION

4
REPRESENTATION OF VECTOR QUANTITIES

• VECTORS ARE REPRESENTED BY AN ARROW

• tip

• tail

Click to Animate
5
THE ARROW

• LENGTH
• THE MAGNITUDE OR SIZE OF THE VECTOR
• THE ARROWS DIRECTION
• IS THE DIRECTION OF THE VECTOR

6
EXAMPLES OF VECTORS

• FORCE (a push or a pull)
• ELECTRIC/MAGNETIC FIELD STRENGTH
• ACCELERATION
• TORQUE twist causing rotation
• DISPLACEMENT not distance
• MOMENTUM possessed by moving mass
• VELOCITY not speed

7
EXAMPLES OF SCALARS

• Mass
• Time
• Distance
• Energy
• Everything else thats not a vector..

These quantities have NO DIRECTION
8
Answer questions 1 3 on the worksheet
9
How to ADD VECTORS

• Take care here
• You Can NOT Add them like regular numbers (called
  Scalars)

10
VECTOR ADDITION (THE TIP-TO-TAIL METHOD) FINDING
THE RESULTANT
Click to Animate

• The SUM or RESULT of Adding 2 Vectors is called

A

B
THE RESULTANT
11
VECTOR ADDITION (THE TIP-TO-TAIL METHOD) FINDING
THE RESULTANT
Click to Animate
A

B
Yeilds
A
B
or
A
B
THE RESULTANT
12
REVIEWING VECTOR ADDITION
Click to Animate

• ADD VECTORS IN ANY ORDER (AB BA)
• IF VECTORS ARE POINTING IN THE SAME DIRECTION ?
  THIS IS REGULAR ALGABRAIC ADDITION

13
REVIEWING VECTOR ADDITION
Click to Animate

• POSITION THE TAIL OF ONE VECTOR TO THE TIP OF THE
  OTHER
• CONNECT FROM THE TAIL OF THE 1ST VECTOR TO THE
  TIP OF THE LAST
• THIS IS THE RESULTANT

14
How About Vectors in Exactly Opposite
DIRECTIONS ?
15
ADDITION CONTINUED
Click to Animate
A
B
A
B
RESULTANT
16
MORE VECTOR ADDITION

• SUPPOSE THE VECTORS FORM A RIGHT ANGLE
• GRAPHICAL SOLUTIONS CAN ALWAYS BE USED BUT
• HERE IS A MATHEMATICAL SOLUTION.
• THIS SOLUTION USES THE PYTHAGOREAN THEORUM
• C2 A2 B2

17
ADDING VECTORS THAT ARE AT RIGHT ANGLES TO EACH
OTHER
Click to Animate
R ?? lbs
B 3 lbs
R2 16 9 25 R 5 lbs
A 4 lbs
BUT R 5 lbs IS ONLY HALF AN ANSWER!! WHY?????
R2 A2 B2 R2 42 32
18
REMEMBER !!!
Click to Animate

• VECTORS HAVE 2 PARTS

MAGNITUDE
AND
DIRECTION !!!
HERES HOW TO FIND THE DIRECTION
19
TRIG FUNCTIONS TO REMEMBER
20
TRIG CALCULATIONS
Click to Animate
R 5 lbs
COS(?) 4/50.8
B 3 lbs
SIN(?) 3/5 0.6
?
A 4 lbs

• USE YOUR CALCULATOR TO FIND THE ANGLE THAT HAS
  THESE VALUES OF SIN OR COS. Could also use TAN

21
AT LAST THE ANGLE (THE VECTORS DIRECTION)
Click to Animate
SIN(X) 0.6 ? ANGLE (X) 37 DEGREES
COS(X) 0.8 ?ANGLE (X) 37 DEGREES
SO, THE OTHER HALF OF OUR ANSWER IS..
22
RESULTANT.
5 lbs 37 degrees NORTH OF EAST

• Not NORTHEAST
• i.e. NE is 45 deg

R 5 lbs
B 3 lbs
? 37 deg
A 4 lbs
23
Answer questions 4 7 on the worksheet
24
SUMMARY
Click to Animate

• A VECTOR IS A DIRECTED QUANTITY THAT HAS BOTH A
  MAGNITUDE AND DIRECTION

• IF THE ANGLE BETWEEN THE VECTORS IS

• 0 deg algebraic addition (MAXIMUM ANS.)

• 180 deg algebraic subtraction (MINIMUM ANS.)

• 90 deg use Pythagorean Theorem to find
  magnitude and trig functions to find the angle

25
THE END OF PART 1 !!!!
RETURN TO BEGINNING

• COULD YOU PASS A QUIZ ON THIS MATERIAL????
• NOW?
• LATER, WITH STUDY?

CONTINUE TO VECTOR MATH
26
Vector Concepts used in Physics Fancy Foot Work

• Imagine you were asked to mark your starting
  place and walk 3 meters North, followed by two
  meters East.
• Could you answer the following
• How far did you walk?
• Where are you relative to your original spot?

27
Fancy Foot Work

• How far did you walk?
• This requires a MAGNITUDE ONLY
• SCALAR QUANTITY called DISTANCE
• 3m 2m 5m
• Where are you relative to your original spot?
• This requires both a MAGNITUDE DIRECTION
• VECTOR QUANTITY called DISPLACEMENT

28
Answer question 8 on the worksheet
29
Fancy Foot Work

• Where are you relative to your original spot?
• This requires both a MAGNITUDE DIRECTION
• VECTOR QUANTITY called DISPLACEMENT

Start
30
Fancy Foot Work

• Magnitude ?
• The Length of the Hypotenuse
• s2(3m)2 (2m)2
• s
• Direction ? East of North
• Pick your Trig function
• ?33.7o, E of N

31
Fancy Foot Work

• NOW, measure the angle from the X axis

By convention, measure all angles
Counterclockwise from the X Axis
?33.7
90 33.7 56.3o
32
Multiplying a Vector by a Scalar

• When you multiply a vector by a scalar, it only
  affects the MAGNITUDE of the vector
• Not the direction
• Example

33
Answer questions 9-11 on the worksheet
34
Vector Components

• Component means part
• A vector can be composed of many parts known as
  components
• Its best to break a vector down into TWO
  perpendicular components. WHY?
• To use Right Triangle Trig

2 perpendicular components
many components
35
Vector Components

• Introducing Vector V
• Vector Vs X-Component is its Projection onto the
  X-axis
• Vector Vs Y-Component is its Projection onto the
  Y-axis

Sub Scripts in Action
Now we have a Right Triangle
Vy
Vx
36
Vector Components

• Given this diagram, find Vs X Y Components

Vx
5
4
Vy
?38.66o
Whats the Magnitude of Vector V?
37
Vector Components

• Now, knowing the magnitude of vector V, verify
  the Vs X Y components using Trig

Here's a check
Vx? Vy?
Vy
38.66o
?
?
38
Vector Components
Important Stuff Gang

• Golden Rules of Vector Components
• 1. If you know the magnitude and direction of
  vector V to be (V,?), then you can find Vx Vy
  by
• VxVcos?
• VyVsin?
• 2. If Vx Vy are known, the magnitude of the
  vector can be found with Pythagorean Theorem

FIND THESE RULES IN YOUR PHYSICS DATA BOOKLET!!
39
WHAT ARE THE X Y COMPONENTS OF VECTOR A?
THESE ARE The VECTOR COMPONENTS OF A
Sin qOPP/HYP Sin qAy/A Ay Asinq
Ax
Cos q ADJ/HYP Cos q Ax/A Ax Acosq
A
Ay
Asinq
Ay
q
Ax
Acosq
40
AN EXAMPLE

• Suppose the magnitude of A 5 and q 37 deg.
  Find the VALUES of the X Y components.

COMPONENTS
5
Ay
3
Asinq
A
Ay
5sin 37
5(0.6)
q
37
Ax
Acosq
5cos37
5(0.8)
4
41
Answer question 12 on the worksheet
42
Vector Components Example of the Golden Rule
TRY IT ON YOUR OWN

• Example Youre a pilot are instructed to go
  around a massive thunderstorm. The control tower
  tell you take a detour follow these 2 paths
• 100 km, 45o 90 km, 10o
• What is the planes displacement from where it
  began its detour?

43
KINEMATICS
Vector Components

• Let V1100 km, 45o V2 90 km, 10o

MAGNITUDE
DIRECTION
44
A SUMMARY

• MAKE A ROUGH SKETCH OF THE VECTORS USING THE
  INFORMATION GIVEN.
• FIND THE X- AND Y- COMPONENT OF ALL OF THE
  VECTORS. (sometimes a table of values is helpful)
• ADD ALL OF THE VECTORS IN THE X-DIRECTION. (check
  the tips and tails of each vector ----vectors
  pointing in the same direction are added
  algebraically in opposite directions-- this is
  algebraic subtraction).
• THIS RESULT WILL GIVE YOU THE X-PART OF THE
  RESULTANT

45

• ADD ALL THE VECTORS IN THE Y-DIRECTION. AGAIN,
  FROM YOUR SKETCH, CHECK THE DIRECTION OF EACH
  Y-VALUE. (ALG.ADD. OR ALG. SUBT.)
• THIS RESULT WILL GIVE YOU THE Y-PART OF THE
  RESULTANT (either pointing up/down or is zero)
• ROUGHLY SKETCH THE RESULTANT
• USING THE PYTHAGOREAN THEOREM FIND THE MAGNITUDE
  OF THE RESULTANT.
• USING THE ARCTAN FORMULA, FIND THE ANGLE THIS
  RESULTANT MAKES WITH THE AXES.
• STATE YOUR ANSWER WITH A MAGNITUDE (including a
  unit) AND THE DIRECTION.

46
MORE EXAMPLES
47
QUESTION 1
A couple on vacation are about to go sight-seeing
in a city where the city blocks are all squares.
They start out at their hotel and tour the city
by walking as follows
1 block East 2 blocks North 3 blocks East 3
blocks South 2 blocks West1 block South 6
blocks East 8 blocks North 8 blocks West. WHAT
IS THEIR DISPLACEMENT? (i.e., WHERE ARE THEY FROM
THEIR HOTEL)?
48
ANSWER 1
USING GRAPH PAPER
1 block East 2 blocks North 3 blocks East 3
blocks South 2 blocks West1 block South 6
blocks East 8 blocks North 8 blocks West.
WHERE ARE THEY FROM THE HOTEL?
THEY ARE 6 BLOCKS NORTH OF THE HOTEL
H
49
ANOTHER METHOD SUM THE COMPONENTS IN THE X AND Y
DIRECTIONS (THEN USE TRIG AS IF IT WAS A SINGLE
VECTOR)
1 block East 2 blocks North 3 blocks East 3
blocks South 2 blocks West1 block South 6
blocks East 8 blocks North 8 blocks West.

N-S ? Y-axis E-W ? X-axis

N() S(-) E() W(-)
2
1
-3
-2
THEY ARE 6 BLOCKS NORTH OF THE HOTEL
8
-1
3
-8
6
6
0
10
-4
-10
10
50
QUESTION 2
RIVER
BOAT
A river flows in the east-west direction with a
current 6 mph eastward. A kayaker (who can
paddle in still water at a maximum rate of 8 mph)
wishes to cross the river in his boat to the
North. If he points the bow of his boat directly
across the river and paddles as hard as he can,
what will be his resultant velocity?
51
ANSWER 2
6
USE PYTHAGOREAN THEOREM
8
RESULTANT VELOCITY
q
R2 62 82 R2 36 64 R2 100 R 10
R 10 mph, 37 deg East of North or 10 mph,
bearing 037 deg.
52
QUESTION 3
BOAT
RIVER
Desired path to the South shore
The kayaker wants to go directly across the river
from the North shore to the South shore, again,
paddling as fast as he can. At what angle should
the kayaker point the bow of his boat so that he
will travel directly across the river? What will
be his resultant velocity?
53
ANSWER 3...
BOAT
RIVER
Desired path to the South shore
q
RESULTANT VELOCITY
VEL.BOAT
8 mph
R
arctan arctan(opp/adj) arc tan(6/5.3)
arctan 1.1 49 deg upstream
q
VEL.RIVER
82 62 Vr2 Vr2 64 - 36 5.3 mph
6 mph
54
QUESTION 4
A
Two forces A and B of 80 and 60 newtons
respectively, act concurrently(at the same point,
at the same time) on point P.
Calculate the resultant force.
B
P
55
ANSWER 4
Pythagorean Theorem for 3-4-5 right triangle
A
R 10 newtons
RESULTANT
80 newtons
arctan (80/60) 53 deg
q
q
B
P
60 newtons